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Complement (set theory)

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From Wikipedia, the free encyclopedia

Set of the elements not in a given subset

A circle filled with red inside a square. The area outside the circle is unfilled. The borders of both the circle and the square are black.

IfA is the area colored red in this image…

An unfilled circle inside a square. The area inside the square not covered by the circle is filled with red. The borders of both the circle and the square are black.

… then the complement ofA is everything else.

Inset theory, thecomplement of asetA, often denoted by $A^{c}$ (orA′),^[1] is the set ofelements not inA.^[2]

When all elements in theuniverse, i.e. all elements under consideration, are considered to bemembers of a given setU, theabsolute complement ofA is the set of elements inU that are not inA.

Therelative complement ofA with respect to a setB, also termed theset difference ofB andA, written $B\setminus A,$ is the set of elements inB that are not inA.

Absolute complement

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Definition

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IfA is a set, then theabsolute complement ofA (or simply thecomplement ofA) is the set of elements not inA (within a larger set that is implicitly defined). In other words, letU be a set that contains all the elements under study; if there is no need to mentionU, either because it has been previously specified, or it is obvious and unique, then the absolute complement ofA is the relative complement ofA inU:^[3] $A^{c}=U\setminus A=\{x\in U:x\notin A\}.$

The absolute complement ofA is usually denoted by $A^{c}$ . Other notations include ${\overline {A}},A',$ ^[2] $\complement _{U}A,{\text{ and }}\complement A.$ ^[4]

Examples

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Assume that the universe is the set ofintegers. IfA is the set of odd numbers, then the complement ofA is the set of even numbers. IfB is the set ofmultiples of 3, then the complement ofB is the set of numberscongruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
Assume that the universe is thestandard 52-card deck. If the setA is the suit of spades, then the complement ofA is theunion of the suits of clubs, diamonds, and hearts. If the setB is the union of the suits of clubs and diamonds, then the complement ofB is the union of the suits of hearts and spades.
When the universe is theuniverse of sets described in formalizedset theory, the absolute complement of a set is generally not itself a set, but rather aproper class. For more info, seeuniversal set.

Properties

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LetA andB be two sets in a universeU. The following identities capture important properties of absolute complements:

De Morgan's laws:^[5]

$\left(A\cup B\right)^{c}=A^{c}\cap B^{c}.$
$\left(A\cap B\right)^{c}=A^{c}\cup B^{c}.$

Complement laws:^[5]

$A\cup A^{c}=U.$
$A\cap A^{c}=\emptyset .$
$\emptyset ^{c}=U.$
$U^{c}=\emptyset .$
${\text{If }}A\subseteq B{\text{, then }}B^{c}\subseteq A^{c}.$
(this follows from the equivalence of a conditional with itscontrapositive).

Involution or double complement law:

$\left(A^{c}\right)^{c}=A.$

Relationships between relative and absolute complements:

$A\setminus B=A\cap B^{c}.$
$(A\setminus B)^{c}=A^{c}\cup B=A^{c}\cup (B\cap A).$

Relationship with a set difference:

$A^{c}\setminus B^{c}=B\setminus A.$

The first two complement laws above show that ifA is a non-empty,proper subset ofU, then{A,A^∁} is apartition ofU.

Relative complement

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Definition

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IfA andB are sets, then therelative complement ofA inB,^[5] also termed theset difference ofB andA,^[6] is the set of elements inB but not inA.

Therelative complement ofA inB: $B\cap A^{c}=B\setminus A$

{\displaystyle B\cap A^{c}=B\setminus A} — Therelative complement ofA inB: $B\cap A^{c}=B\setminus A$

The relative complement ofA inB is denoted $B\setminus A$ according to theISO 31-11 standard. It is sometimes written $B-A,$ but this notation is ambiguous, as in some contexts (for example,Minkowski set operations infunctional analysis) it can be interpreted as the set of all elements $b-a,$ whereb is taken fromB anda fromA.

Formally: $B\setminus A=\{x\in B:x\notin A\}.$

Examples

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$\{1,2,3\}\setminus \{2,3,4\}=\{1\}.$
$\{2,3,4\}\setminus \{1,2,3\}=\{4\}.$
If $\mathbb {R}$ is the set ofreal numbers and $\mathbb {Q}$ is the set ofrational numbers, then $\mathbb {R} \setminus \mathbb {Q}$ is the set ofirrational numbers.

Properties

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LetA,B, andC be three sets in a universeU. The followingidentities capture notable properties of relative complements:

$C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B).$
$C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B).$
$C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B),$
with the important special case $C\setminus (C\setminus A)=(C\cap A)$ demonstrating that intersection can be expressed using only the relative complement operation.
$(B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).$
$(B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).$
$A\setminus A=\emptyset .$
$\emptyset \setminus A=\emptyset .$
$A\setminus \emptyset =A.$
$A\setminus U=\emptyset .$
If $A\subset B$ , then $C\setminus A\supset C\setminus B$ .
$A\supseteq B\setminus C$ is equivalent to $C\supseteq B\setminus A$ .

Complementary relation

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Abinary relation $R {\displaystyle R}$ is defined as a subset of aproduct of sets $X\times Y.$ Thecomplementary relation ${\bar {R}}$ is the set complement of $R {\displaystyle R}$ in $X\times Y.$ The complement of relation $R {\displaystyle R}$ can be written ${\bar {R}}\ =\ (X\times Y)\setminus R.$ Here, $R {\displaystyle R}$ is often viewed as alogical matrix with rows representing the elements of $X, {\displaystyle X,}$ and columns elements of $Y . {\displaystyle Y.}$ The truth of $a R b {\displaystyle aRb}$ corresponds to 1 in row $a, {\displaystyle a,}$ column $b . {\displaystyle b.}$ Producing the complementary relation to $R {\displaystyle R}$ then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together withcomposition of relations andconverse relations, complementary relations and thealgebra of sets are the elementaryoperations of thecalculus of relations.

LaTeX notation

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In theLaTeX typesetting language, the command\setminus^[7] is usually used for rendering a set difference symbol, which is similar to abackslash symbol. When rendered, the\setminus command looks identical to\backslash, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence\mathbin{\backslash}. A variant\smallsetminus is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol $\complement$ (as opposed to $C {\displaystyle C}$ ) is produced by\complement. (It corresponds to the Unicode symbolU+2201 ∁COMPLEMENT.)

Notes

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^"Complement and Set Difference".web.mnstate.edu. Retrieved2020-09-04.
^^a ^b"Complement (set) Definition (Illustrated Mathematics Dictionary)".www.mathsisfun.com. Retrieved2020-09-04.
^The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
^Bourbaki 1970, p. E II.6.
^^a ^b ^cHalmos 1960, p. 17.
^Devlin 1979, p. 6.
^[1]Archived 2022-03-05 at theWayback Machine The Comprehensive LaTeX Symbol List

References

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Bourbaki, N. (1970).Théorie des ensembles (in French). Paris: Hermann.ISBN 978-3-540-34034-8.
Devlin, Keith J. (1979).Fundamentals of contemporary set theory. Universitext.Springer.ISBN 0-387-90441-7.Zbl 0407.04003.
Halmos, Paul R. (1960).Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company.ISBN 9780442030643.Zbl 0087.04403.{{cite book}}:ISBN / Date incompatibility (help)

External links

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